If the fourth term in the binomial expansion of

Question:

If the fourth term in the binomial expansion of

$\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}$ is equal to 200 , and $x>1$,

then the value of $\mathrm{x}$ is :

  1. $10^{3}$

  2. 100

  3. $10^{4}$

  4. 10


Correct Option: , 4

Solution:

$200={ }^{6} C_{3}\left(x^{\frac{1}{x+\log _{10} x}}\right)^{\frac{3}{2}} \times x^{\frac{1}{4}}$

$\Rightarrow 10=x^{\frac{3}{2\left(1+\log _{10} x\right)} \cdot \frac{1}{4}}$

$\Rightarrow 1=\left(\frac{3}{2(1+t)}+\frac{1}{4}\right) t$

where $t=\log _{10} x$

$\Rightarrow \mathrm{t}^{2}+3 \mathrm{t}-4=0$

$\Rightarrow \mathrm{t}=1,-4$

$\Rightarrow \mathrm{x}=10,10^{-4}$

$\Rightarrow \mathrm{x}=10($ As $\mathrm{x}>1)$

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